Breathing multichimera states in nonlocally coupled phase oscillators

Abstract

Chimera states for the one-dimensional array of nonlocally coupled
phase oscillators in the continuous limit are assumed to be steady
states in most studies, but a few studies report the existence of
breathing chimera states.
In this paper, we focus on multichimera states with two coherent and
incoherent regions, and numerically demonstrate that breathing
multichimera states, whose global order parameter oscillates
temporally, can appear.
Moreover, we show that the system exhibits a Hopf bifurcation from a
steady multichimera to a breathing one by the linear stability analysis
for the steady multichimera.

Chimera states for the one-dimensional array of phase oscillators in the
continuous limit N→∞, where N is the number of oscillators,
are mostly assumed to be macroscopically time-independent steady states
in various studies, e.g. the self-consistent equation of the local mean
field Kuramoto and Battogtokh (2002); Abrams and Strogatz (2004, 2006); Sethia et al. (2008) and the linear stability analysis for the
chimera state Omel’chenko (2013); Xie et al. (2014); Smirnov et al. (2017).
However, Abrams et al. Abrams et al. (2008) discovered
breathing chimeras, whose global order parameter of a population
oscillates temporally, for two interacting populations of globally
coupled phase oscillators, and posed a question whether such breathing
chimeras exist in the case of the one-dimensional arrays.
As the answer of this question, Laing Laing (2009)
demonstrated that there also appear breathing chimeras in the
one-dimensional system by introducing phase lag parameter heterogeneity.

In this paper, we focus on chimera states, especially multichimera
states with two coherent and incoherent regions, in one-dimensional
nonlocally coupled phase oscillators.
Moreover, it is demonstrated that breathing chimeras can appear even in
homogeneous systems without introducing parameter heterogeneity.
By numerical simulations, we observe that the appearance of breathing
chimeras depends on the coupling kernel function.
Then we show that the system exhibits a Hopf bifurcation from a steady
chimera to a breathing one by the linear stability analysis for the
steady chimera.

Ii Model

We consider the system of nonlocally coupled phase oscillators obeying

˙θ(x,t)=ω−∫π−πdyG(x−y)sin(θ(x,t)−θ(y,t)+α)

(1)

with 2π-periodic phase θ(x,t) on the one-dimensional space
x∈[−π,π] under the periodic boundary condition.
The coupling between oscillators is assumed to be the sine function with
phase lag parameter αSakaguchi and Kuramoto (1986), and the
natural frequency ω can be set to zero without loss of
generality.
The coupling kernel function G(x) is generally an even real function
described as

with 0<r≤1, where r denotes the coupling range.
For numerical simulations of Eq. (1), we need to
discretize x into xj:=−π+2πj/N (j=1,⋯,N).
Then Eq. (1) with the step kernel
Eq. (3) are rewritten as

˙θj(t)=ω−12Rj+R∑k=j−Rsin(θj(t)−θk(t)+α),

(4)

where θj(t):=θ(xj,t) and R:=rN/2.
In this paper, we use the fourth-order Runge-Kutta method with time
interval Δt=0.01 for all numerical simulations.
Note that chimera states for Eq. (4) with finite N
are often unstable and merely transient for especially small
NWolfrum et al. (2011); Wolfrum and Omel’chenko (2011); Rosin et al. (2014); Suda and Okuda (2015).

For Eq. (4), there appear various types of chimera
states Maistrenko et al. (2014), including multichimera states with two or
more coherent and incoherent regions.
A typical multichimera state obtained by numerical simulation is shown
in Fig. 1.
Fig. 1(a) shows the snapshot of the phase θ(x,t),
and Fig. 1(b) shows the profile of the average frequency
⟨˙θ(x)⟩:=∫T0dt′˙θ(x,t′)/T
with the measurement time T.
When we refer to time averaged quantities ⟨⋅⟩ in our
numerical simulations, we set the measurement time to T=2000 and
measured those quantities after the transient time 2000.
The multichimera state in Fig. 1 has two coherent and
incoherent regions, which we call 2-multichimera below.
As seen from the phase snapshot in Fig. 1(a), the two
coherent regions are separated from each other by the phase almost
exactly π, which is a remarkable feature of the 2-multichimera
different from mere two neighboring chimeras.
The stability region of 2-multichimera for Eq. (4)
obtained by the numerical simulation with N=100000 is shown in
Fig. 2.
This result is consistent with the phase diagram
in Maistrenko et al. (2014), except for the existence of breathing
multichimeras.

Figure 1:
Multichimera state with two coherent and incoherent regions for
Eq. (4) with N=10000, α=1.480, and
r=0.360.
The top panel (a) shows the snapshot of the phase θ(x,t), and
the bottom panel (b) shows the profile of the average frequency
⟨˙θ(x)⟩ with T=2000.
Figure 2:
Stability region of 2-multichimera for Eq. (4)
obtained by the numerical simulation with N=100000.
Black circles denote the parameter values of
Fig. 3.
The blue line denotes the Hopf bifurcation points obtained by the
linear stability analysis for the steady 2-multichimera with fixed
α in Sec. IV, but we could not determine those
points for α close to π/2 (dashed line).

It is mostly assumed that the chimera state for
Eq. (1) is a steady state in the rotating frame with a
frequency Ω.
This means that the local mean field

Y(x,t):=∫π−πdyG(x−y)eiθ(y,t),

(5)

acting on the oscillator located in point x, takes the form
Y(x,t)=~Y(x)eiΩt.
Then the global order parameter

Z(t):=12π∫π−πdyeiθ(y,t),

(6)

also takes the form Z(t)=~ZeiΩt.
Here, |Z(t)| denotes the synchronization degree of all oscillators,
that is, all oscillators are completely synchronized in phase for
|Z(t)|=1 and otherwise for 0≤|Z(t)|<1.
In the case of the steady 2-multichimera as in Fig. 1,
|Z(t)| should vanish in the continuous limit N→∞, but we
found that |Z(t)| can oscillate periodically at appropriate parameters
(r,α) and sufficiently large N.
Fig. 3 shows the time evolution of |Z(t)| for
2-multichimeras with N=100000.
The blue line (α=1.480 and r=0.360) exhibits a clear periodic
oscillation, while the red line (α=1.480 and r=0.440) merely
exhibits a small fluctuation around 0.
We call the former state breathing multichimera, while we regard the
latter as steady multichimera.

Figure 3:
Time evolution of the global order parameter |Z(t)| for a
2-multichimera for Eq. (4) with N=100000 and
α=1.480.
The 2-multichimera is breathing for r=0.360 (blue line), while steady
for r=0.440 (red line).
Dashed lines correspond to the times t in Fig. 4.

The detailed periodic behavior of the breathing 2-multichimera can be
confirmed as the periodic oscillation of |Y(x,t)| as shown in
Fig. 4.
|Y(x,t)| takes a bimodal form, where the positions of the peaks
correspond to each center of the coherent regions.
Within a period of the global order parameter |Z(t)| approximately
corresponding to t=12∼24 in Fig. 3, |Y(x,t)|
experiences the variation in the half of its period, and within the next
period of |Z(t)|, |Y(x,t)| completes its whole period.
Therefore, the period of |Y(x,t)| is double of that of |Z(t)|.
In the simulation of Fig. 4, the angular frequency of
|Y(x,t)| is calculated as about 0.270, which is compared with the
result of the linear stability analysis in Sec. IV.

Figure 4:
Snapshot of the local mean field |Y(x,t)| for the breathing
2-multichimera for Eq. (4) with N=100000,
α=1.480, and r=0.360.
The top panel (a) shows the global view of the snapshot, and the bottom
panel (b) shows the upper enlarged view.
The red (t=12) and green lines (t=24) approximately correspond to
peaks of the global order parameter |Z(t)|, while the blue line
(t=18) approximately corresponds to a valley, as shown in
Fig. 3.

We can distinguish between steady and breathing 2-multichimeras by
studying the time evolution of |Z(t)|, but that is difficult for small
N because large fluctuation in |Z(t)| is unavoidable.
To distinguish these clearly, we needed 10000 oscillators at least in
our numerical simulation.
Though such breathing properties of the standard chimera states in
one-dimensional phase oscillators systems is observed by introducing
phase lag parameter heterogeneity Laing (2009), we note that
the present breathing 2-multichimera does not require such
heterogeneity.
In this paper, we focus on this breathing 2-miltichimera, and study the
bifurcation mechanism from the steady 2-multichimera.

Iii Steady 2-multichimera

First, we study basic properties of steady 2-multichimeras.
We first rewrite Eq. (1) as

and obtain Y(x,t)=∫π−πdyG(x−y)z(y,t).
|z(x,t)| denotes the synchronization degree of oscillators around
point x at time t, similarly to the global order parameter |Z(t)|.
For |z(x,t)|=1, the oscillators in the neighborhood of x are
completely synchronized in phase.
Otherwise, when their phases are scattered, we obtain 0≤|z(x,t)|<1.
Therefore, we can identify |z(x,t)|=1 and 0≤|z(x,t)|<1 as the
coherent and incoherent regions for chimera states, respectively.

Following the method in Pikovsky and Rosenblum (2008); Wolfrum et al. (2011),
we can obtain the evolution equation of z(x,t) as

˙z(x,t)=iωz(x,t)+12e−iαY(x,t)−12eiαz2(x,t)Y∗(x,t),

(9)

using Eq. (7), where the symbol ∗ denotes the
complex conjugate.
Eq. (9) can also be obtained by another
method Laing (2009); Omel’chenko (2013) using the
Ott-Antonsen ansatz Ott and Antonsen (2008, 2009).
Substituting the steady solution z(x,t)=~z(x)eiΩt
into Eq. (9) and integrating it, we can obtain the
self-consistent equation

where R(x)eiΘ(x):=~Y(x)=Y(x,t)e−iΩt and
Δ:=ω−Ω.
These equations correspond to the self-consistent equation derived by
Kuramoto and Battogtokh Kuramoto and Battogtokh (2002).
Eqs. (10) and (11) are composed by two
equations given by the real and imaginary parts, but have three real
unknowns R(x), Θ(x) and Δ.
Therefore, we need to add the third condition to solve them.
The third condition can be obtained from the fact that
Eqs. (10) and (11) are invariant under
any rotation
Θ(x)→Θ(x)+Θ0Abrams and Strogatz (2004, 2006); Sethia et al. (2008); Xie et al. (2014).
Based on the above, we have chosen the condition

Θ(−π)=−π2.

(12)

Eqs. (10) and (11) under
Eq. (12) can be solved by the following iteration
procedure numerically Abrams and Strogatz (2006); Sethia et al. (2008).
First, we prepare an initial function ~Y(x) i.e. R(x) and
Θ(x), and obtain Δ satisfying Eq. (12) from
Eqs. (10) and (11) by Newton’s method
with respect to Δ.
Second, substituting ~Y(x) and Δ into the right hand
side of Eq. (10), we generate a new ~Y(x) from
the left hand side.
Third, we obtain a new Δ satisfying Eq. (12) again
by Newton’s method using the new ~Y(x).
It only remains to repeat the second and third steps until both
~Y(x) and Δ converge.

According to Omel’chenko (2013), it is analytically proved that
2-multichimeras exist under the condition g1≠0 for the coupling
kernel Eq. (2) and the local mean field ~Y(x) of
a steady 2-multichimera is given by an even function

~Y(x)=∞∑m=1C2m−1cos((2m−1)x),

(13)

where C2m−1∈C.
Using Eq. (13), h(x) turns out to be an even function
because R(x)=|~Y(x)| is also even.
Let a set of ~Y(x) satisfying Eq. (13) and
Δ be a solution of Eqs. (10)
and (11).
Substituting Eqs. (2) and (13) into
Eqs. (10) and (11) and eliminating the
terms whose integrands are odd functions of y, we obtain

~Y(x)

=

2ie−iα∞∑k=0gkcos(kx)∞∑m=1C2m−1

(14)

×

∫π0dycos(ky)cos((2m−1)y)h(y).

Changing the variable as y′=y−π/2 in the integration in
Eq. (14), the function h(y′+π/2) in the integrand is
an even function of y′ because of Eq. (13).
We again eliminate the terms whose integrands are odd functions of y′,
and obtain

~Y(x)=2ie−iα∞∑l=1g2l−1cos((2l−1)x)∞∑m=1C2m−1(−1)l+m

×∫π2−π2dy′sin((2l−1)y′)sin((2m−1)y′)h(y′+π2).

(15)

From the above, for ~Y(x) and Δ of a steady
2-multichimera satisfying Eqs. (10)
and (11), we can finally obtain

~Y(x)=ie−iα∞∑l=1g2l−1∞∑m=1C2m−1Almcos((2l−1)x),

(16)

where Alm is a complex constant.
Eq. (16) shows that ~Y(x) and Δ of a
steady 2-multichimera depends only on the odd harmonic coefficients
g2m−1, not on the even harmonic coefficients g2m of the
coupling kernel G(x).
This is because we recover Eq. (16) even when we
substitute the identical set of ~Y(x) and Δ into
Eqs. (10) and (11) with another coupling
kernel, for example,

Godd(x)=∞∑m=1g2m−1cos((2m−1)x),

(17)

having the same set of odd harmonic coefficients g2m−1.
Therefore, if there appears a steady 2-multichimera for
Eq. (1) with Godd(x), it has the same local
mean field as for the original coupling kernel.
This property of the coupling kernel Godd(x) is useful in the
linear stability analysis for steady 2-multichimeras mentioned in
Sec. IV.

To illustrate the above property for our step kernel
Eq. (3), we performed a numerical simulation using a
new coupling kernel Godd(x) with the same set of odd harmonic
coefficients g2m−1 as for Eq. (3).
Godd(x) can also be obtained as
Godd(x)=[G(x)−G(x−π)]/2, which we used in the numerical
simulation instead of the Fourier expansion in Eq. (17).
Then, also for this Godd(x), we observed steady
2-multichimeras.
Fig. 5 shows the time-averaged local mean fields
⟨~Y(x)⟩ of the steady 2-multichimeras using the
step kernel G(x) and the corresponding Godd(x).
They are clearly identical, and also agree with the numerical solution
~Y(x) to the self-consistent equation Eqs. (10)
and (11) with Godd(x).
Note that the solid lines in Fig. 5 are obtained for
Godd(x), not for G(x).
We tried to solve Eqs. (10) and (11)
with the step kernel G(x) numerically, but we could not obtain a
steady solution ~Y(x) of 2-multichimera, because ~Y(x)
converged to another solution corresponding to a standard chimera state
with one coherent and one incoherent region, under any initial
conditions.

Figure 5:
Local mean field ~Y(x) of a steady 2-multichimera.
The top panel (a) shows the amplitude R(x), and the bottom panel (b)
shows the argument Θ(x).
Open circles denote the time-averaged local mean field
⟨~Y(x)⟩ for Eq. (1) with the step
kernel G(x), namely Eq. (4), with N=100000,
α=1.480, and r=0.440, and open squares denote
⟨~Y(x)⟩ for Godd(x) with the same
parameters.
Note that those are plotted once every 2000 oscillators.
The solid line denotes the numerical solution ~Y(x) to the
self-consistent equation Eqs. (10)
and (11) with Godd(x).

2-multichimeras can also appear for Eq. (1) with
other Godd(x) e.g.
Godd(x)=g1cos(x)Xie et al. (2014),
Godd(x)=g1cos(x)+g3cos(3x) as shown in
Fig. 6, and so on.
In our numerical simulations for various Godd(x) systems, we
found an interesting property common to 2-multichimeras for
Godd(x), which is an exact relationship between the phase
θ(x,t) as

|θ(x,t)−θ(x−π,t)|=π,

(18)

on any point x.
In fact, from Eq. (7) with any Godd(x), we
obtain ˙θ(x,t)−˙θ(x−π,t)=0 for a solution
Eq. (18), by using the relation Y(x,t)=−Y(x−π,t)
satisfied at any time.
This implies that Eq. (18) can be a solution to
Eq. (1) with Godd(x) whether stable or not, but
our simulations show that the system with Godd(x) always
converges to the solution Eq. (18).
For the kernel other than Godd(x), this property
Eq. (18) is not exact, but seems to be satisfied only
in the meaning of average, as seen in Fig. 1(a).

Figure 6:
Snapshot of a 2-multichimera for Eq. (1) with
Godd(x)=g1cos(x)+g3cos(3x) with N=10000,
α=1.500, g1=1, and g3=−0.0916.
The phase θ(x,t) on any point x satisfies
Eq. (18).

Iv Breathing 2-multichimera

As described above, 2-multichimeras for Eq. (1) with the
step kernel G(x) do not satisfy Eq. (18).
In our simulations, however, we often observed breathing 2-multichimeras
instead of steady 2-multichimeras, as shown in
Fig. 3.
This breathing 2-multichimera is characterized by the global order
parameter |Z(t)| oscillating periodically.
Therefore, 2-multichimeras for Godd(x) satisfying
Eq. (18) cannot breathe, because the global order
parameter of Eq. (18) exactly vanishes.

It is known that the breathing chimeras in the other
studies Abrams et al. (2008); Laing (2009); Panaggio et al. (2016) branch via Hopf bifurcation from stable steady
chimeras.
If the present breathing 2-multichimera also branches via Hopf
bifurcation, an unstable steady 2-multichimera should exist in the
neighborhood of the bifurcation point.
The local mean field of this unstable steady 2-multichimera should be a
solution to the self-consistent equation Eqs. (10)
and (11), and identical with that of the steady
2-multichimera for the Godd(x) system.

In order to investigate this bifurcation, we analyze the linear
stability of steady 2-multichimeras.
Substituting z(x,t)=(~z(x)+v(x,t))eiΩt with the
steady solution ~z(x)eiΩt and a small perturbation
v(x,t) into Eq. (9), we obtain the linear-order evolution
equation of v(x,t) as

˙v(x,t)=g(x)~z(x)+12e−iαV(x,t)−12eiα~z2(x)V∗(x,t),

(19)

g(x):=⎧⎪⎨⎪⎩i√Δ2−R(x)2(Δ>R(x))−√R(x)2−Δ2(Δ≤R(x)),

(20)

V(x,t):=∫π−πdyG(x−y)v(y,t),

(21)

where Δ=ω−Ω.
We rewrite Eqs. (19)-(21) as
˙v=Lv using
v(x,t)=(Rev(x,t),Imv(x,t))T and solve the
eigenvalue problem of L.
According to Omel’chenko (2013); Xie et al. (2014), the
spectrum of L consists of the essential spectrum and the point
spectrum.
In the present case, the essential spectrum is given by g(x)
consisting of pure imaginary and negative real eigenvalues, which
correspond to incoherent and coherent regions, respectively.
Therefore, the stability of steady 2-multichimeras should be
determined only by the point spectrum.

If the number of nonzero gk in Eq. (2) is finite, we
may solve the eigenvalue problem of a finite size matrix to obtain the
point spectrum Omel’chenko (2013); Xie et al. (2014).
However, the step kernel Eq. (3) has infinite numbers
of nonzero gk.
Therefore, we discretize the space coordinate
x→xj=−π+2πj/M (j=1,⋯,M), and compute all eigenvalues
λ by solving the eigenvalue problem of 2M×2M matrix
Ld such that ˙vd=Ldvd with
vd(t)=(⋯,Rev(xj,t),Imv(xj,t),⋯)TSmirnov et al. (2017).
In order to solve this problem, we first need to prepare
~Y(x) and Δ of the steady 2-multichimera for
Eq. (4) numerically, but could not obtain them by
solving the self-consistent equation Eqs. (10)
and (11) with the step kernel G(x) because
~Y(x) converged to another solution by the iteration procedure,
as mentioned in the section III.
We accordingly used ~Y(x) and Δ of
Eqs. (10) and (11) with the
corresponding Godd(x), instead of G(x).
Fig. 7(a) shows all the eigenvalues λ of
Ld with M=5000, α=1.480, and r=0.360 on the complex
plane.
As seen from the figure, we have some eigenvalues with positive real
part, because the steady 2-multichimera is unstable and changes into a
breathing one at these parameters.
We can regard those eigenvalues as roughly separating into two groups.
The group 1 consists of some eigenvalues around the real axis, and the
group 2 consists of others around the imaginary values about 0.270 and
their complex conjugate, as shown in Fig. 7(c).

Figure 7:
Complex eigenvalues λ of Ld with M=5000 and
α=1.480 using ~Y(x) with N=200000.
The top panel (a) shows all eigenvalues for the unstable steady
2-multichimera that changes to a breathing one (r=0.360), and the
middle panel (b) shows those for the stable steady 2-multichimera
(r=0.440).
The bottom panel (c) shows the enlarged view of panel (a) and (b)
denoted by the blue and red points, respectively.
The dashed lines in each panel are drawn only for reference.

Even though we can observe the eigenvalues with positive real part, we
cannot easily tell whether they belong to the point spectrum or a
fluctuation of the essential spectrum caused by finite discretization.
If an eigenvalue with positive real part belongs to such a fluctuation,
its real part should go to 0 in the continuous limit M→∞, while
an eigenvalue in the point spectrum keeps positive real part in that
limit.
We computed the eigenvalues of Ld with various M and found their
limiting behaviors as M is increased, as shown in
Fig. 8.

From Fig. 8, we can see that the maximum value of
the real parts of the eigenvalues in the group 1 tends to go to 0, while
that value in the group 2 converges to a positive constant.
Therefore, it turns out that at least a pair of the complex conjugate
eigenvalues in the group 2 belongs to the point spectrum, while the
eigenvalues in the group 1 belong to the fluctuation of the essential
spectrum by finite discretization.
At the other parameters where the steady 2-multichimera is stable, the
point spectrum contains only the eigenvalues with negative real part, as
shown in Fig. 7(b).

Figure 8:
Transition of the positive real parts of the eigenvalues of Ld for
an unstable steady 2-multichimera (α=1.480 and r=0.360) as
increasing M.
Open circles denote the maximum values of the real parts of the
eigenvalues in the group 1, and open triangles denote those in the
group 2.
The data for the group 1 are fitted linearly (dashed line) in the
log-log plot, and go to 0 as increasing M.
In contrast, the data for the group 2 converge to a positive constant
1.175×10−2 (solid line).

Fig. 9 shows that a Hopf bifurcation from a steady
2-multichimera to a breathing one occurs for α=1.480 denoted by
the black solid line in Fig. 2.
The Hopf bifurcation points for α=1.480 and other values are
shown as the blue line in Fig. 2.
However, we could not determine the bifurcation points for α
close to π/2, because it is difficult to distinguish the point
spectrum whose real parts are almost 0 for those α.
We note that the absolute values of the imaginary parts of the point
spectrum as shown in Fig. 7(a) are nearly equal to
the angular frequency of the local mean field |Y(x,t)| as shown in
Fig. 4, which is calculated as about 0.270.
This result agrees with the occurrence of a Hopf bifurcation.

Figure 9:
Hopf bifurcation for fixed α=1.480.
Each point shows the eigenvalue with the positive imaginary part and
the maximum absolute value of the real part in the point spectrum for
r=0.440 (red), r=0.420 (orange), r=0.380 (green), and r=0.360
(blue), and corresponds to the black solid line in
Fig. 2.
The point for r=0.400 is omitted in the figure, because we could not
distinguish the point spectrum from other eigenvalues for r=0.400
that is very close to the Hopf bifurcation point.
The dashed line is the imaginary axis.

V Summary

We studied 2-multichimera with two coherent and incoherent regions in
one-dimensional nonlocally coupled phase oscillators described as
Eq. (1).
First, we proved analytically that the local mean field ~Y(x)
of steady 2-multichimeras depends on only odd harmonic coefficients
g2m−1 of the coupling kernel function G(x).
This implies that if G(x) has the same set of the odd harmonic
coefficients, ~Y(x) of steady 2-multichimeras are common to all
those G(x) systems for the same parameters.
We could actually apply ~Y(x) of the Godd(x) system to
the linear stability analysis of steady 2-multichimeras for the G(x)
system, even though we could not obtain ~Y(x) of steady
2-multichimeras for the G(x) system.
The method used in this paper is based on the fact that ~Y(x)
of steady 2-multichimeras is characterized by only odd harmonic
components, namely Eq. (13).
We expect that a similar method is applied to other steady multichimera
states, because their local mean fields are also characterized by a set
of specific harmonic components Omel’chenko (2013).

Next, we numerically found that breathing 2-multichimeras with
oscillatory global order parameter |Z(t)| appear for
Eq. (1) with the step kernel Eq. (3)
without introducing parameter heterogeneity Laing (2009).
Moreover, we clarified that the system exhibits a Hopf bifurcation from
a steady 2-multichimera to a breathing one by the linear stability
analysis for the steady 2-multichimera.
In contrast to the G(x) system, 2-multichimeras for Godd(x)
cannot breathe, because the system converges to the solution
Eq. (18) with vanishing |Z(t)|.
Therefore, it is inferred that the coupling kernel is an important
factor for the appearance of breathing chimeras in the one-dimensional
system.
It may be interesting to find other breathing chimeras by using the
appropriate coupling kernel similarly, but it is an open problem.